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Numbers n such that the sum of the prime distinct divisors of n^2+1 equals 2 times the difference between the largest and the smallest prime divisor.
2

%I #21 Jun 30 2023 03:54:35

%S 447,2042,4942,8673,17232,18321,38232,52953,54468,54974,55174,57229,

%T 66567,71132,83071,101499,113667,121206,133047,173932,297907,325286,

%U 430353,447131,656079,702969,842151,937313,1061846,1173886,1613346,1721094,1754679,1759310

%N Numbers n such that the sum of the prime distinct divisors of n^2+1 equals 2 times the difference between the largest and the smallest prime divisor.

%H Amiram Eldar, <a href="/A200071/b200071.txt">Table of n, a(n) for n = 1..500</a>

%e 447 is a term because the distinct prime divisors of 447^2 + 1 are 2, 5, 13, 29, 53 and their sum, 2 + 5 + 13 + 29 + 53 = 102, equals 2*(53 - 2).

%t Select[Range[1800000],Plus@@(pl=First/@FactorInteger[#^2+1])/2==pl[[-1]]-pl[[1]]&]

%t spddQ[n_]:=Module[{fi=FactorInteger[n^2+1][[All,1]]},Total[fi] == 2*(Last[ fi]-First[fi])]; Select[Range[176*10^4],spddQ] (* _Harvey P. Dale_, Jan 12 2019 *)

%Y Cf. A200070.

%K nonn

%O 1,1

%A _Michel Lagneau_, Nov 13 2011